{"paper":{"title":"Asymptotic Balance and Structural Rigidity in the Riemann Zeta Function","arxiv_id":"2609.28529","primary_cat":"math.GM","submitted_at":"2026-09-22T17:43:38+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.86,"id":"s0","own":true,"role":"contribution","text":"We establish a structural rigidity principle for the nontrivial zeros of the Riemann zeta function by analyzing the asymptotic interaction between discrete Dirichlet oscillations and their continuous analytic envelope.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.17,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.14,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.86,"interpretation":0.0}},{"i":1,"p":0.95,"id":"s1","own":true,"role":"method","text":"The starting point is a renormalized discrepancy operator, the C-transformation, which isolates the finite structural difference between the Dirichlet series and its integral approximation.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.77,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.95,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.05,"interpretation":0.0}},{"i":2,"p":0.93,"id":"s2","own":true,"role":"method","text":"Applied to the kernel x^(-s), this operator yields a decomposition of the zeta function into three parts: an oscillatory Dirichlet component X_N(s), an explicit analytic growth envelope Y_N(s), and a polynomially decaying remainder R_N(s).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.35,"role_probabilities":{"gap":0.0,"method":0.93,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":3,"p":0.81,"id":"s3","own":true,"role":"result","text":"We show that the vanishing condition zeta(s)=0 forces an asymptotic compatibility between the two dominant components.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.81,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.1,"interpretation":0.06}},{"i":4,"p":0.44,"id":"s4","own":true,"role":"result","text":"This relation is formalized in a Bridge Lemma, which proves that zeros require asymptotic equality of the squared magnitudes |X_N(s)|^2 and |Y_N(s)|^2.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.83,"role_probabilities":{"gap":0.0,"method":0.27,"result":0.44,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.29,"interpretation":0.0}},{"i":5,"p":0.96,"id":"s5","own":true,"role":"result","text":"A detailed asymptotic analysis then reveals that the ratio of these magnitudes exhibits three distinct regimes depending on the real part sigma=Re(s): divergence for sigma>1/2, collapse for sigma<1/2, and exact balance for sigma=1/2.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.07,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.96,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":6,"p":0.92,"id":"s6","own":true,"role":"interpretation","text":"This incompatibility of growth regimes produces a rigidity mechanism: the asymptotic balance required by the zero condition can occur only on the critical line.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.83,"main_claim_p":0.14,"quantitative":false,"quantitative_p":0.17,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.08,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.92}},{"i":7,"p":1.0,"id":"s7","own":true,"role":"result","text":"Consequently, every nontrivial zero must satisfy Re(s)=1/2.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.94,"quantitative":true,"quantitative_p":0.89,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":8,"p":1.0,"id":"s8","own":true,"role":"interpretation","text":"The result provides a deterministic explanation for the location of the zeros, showing that the critical line emerges from an intrinsic structural constraint of the Dirichlet representation of the zeta function.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.2,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":1.0}}]},"compiled":{"main_claim":{"p":0.99,"text":"Consequently, every nontrivial zero must satisfy Re(s)=1/2.","location":"abstract 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